Expand Using Pascal's Triangle (4a-1)^3

${(4 a - 1)}^{3}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

The triangle can be used to calculate the coefficients of the expansion of ${(a + b)}^{n}$ by taking the exponent $n$ and adding $1$ . The coefficients will correspond with line $n + 1$ of the triangle. For ${(4 a - 1)}^{3}$ , $n = 3$ so the coefficients of the expansion will correspond with line $4$ .

Step 2

The expansion follows the rule ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are $1 - 3 - 3 - 1$ .

$1 a^{3} b^{0} + 3 a^{2} b + 3 a b^{2} + 1 a^{0} b^{3}$

Step 3

Substitute the actual values of $a$ $4 a$ and $b$ $- 1$ into the expression.

$1 {(4 a)}^{3} {(- 1)}^{0} + 3 {(4 a)}^{2} {(- 1)}^{1} + 3 {(4 a)}^{1} {(- 1)}^{2} + 1 {(4 a)}^{0} {(- 1)}^{3}$

Step 4

Simplify each term.

Multiply ${(4 a)}^{3}$ by $1$ .

${(4 a)}^{3} {(- 1)}^{0} + 3 {(4 a)}^{2} {(- 1)}^{1} + 3 {(4 a)}^{1} {(- 1)}^{2} + 1 {(4 a)}^{0} {(- 1)}^{3}$

Apply the product rule to $4 a$ .

$4^{3} a^{3} {(- 1)}^{0} + 3 {(4 a)}^{2} {(- 1)}^{1} + 3 {(4 a)}^{1} {(- 1)}^{2} + 1 {(4 a)}^{0} {(- 1)}^{3}$

Raise $4$ to the power of $3$ .

$64 a^{3} {(- 1)}^{0} + 3 {(4 a)}^{2} {(- 1)}^{1} + 3 {(4 a)}^{1} {(- 1)}^{2} + 1 {(4 a)}^{0} {(- 1)}^{3}$

Anything raised to $0$ is $1$ .

$64 a^{3} \cdot 1 + 3 {(4 a)}^{2} {(- 1)}^{1} + 3 {(4 a)}^{1} {(- 1)}^{2} + 1 {(4 a)}^{0} {(- 1)}^{3}$

Multiply $64$ by $1$ .

$64 a^{3} + 3 {(4 a)}^{2} {(- 1)}^{1} + 3 {(4 a)}^{1} {(- 1)}^{2} + 1 {(4 a)}^{0} {(- 1)}^{3}$

Apply the product rule to $4 a$ .

$64 a^{3} + 3 (4^{2} a^{2}) {(- 1)}^{1} + 3 {(4 a)}^{1} {(- 1)}^{2} + 1 {(4 a)}^{0} {(- 1)}^{3}$

Raise $4$ to the power of $2$ .

$64 a^{3} + 3 (16 a^{2}) {(- 1)}^{1} + 3 {(4 a)}^{1} {(- 1)}^{2} + 1 {(4 a)}^{0} {(- 1)}^{3}$

Multiply $16$ by $3$ .

$64 a^{3} + 48 a^{2} {(- 1)}^{1} + 3 {(4 a)}^{1} {(- 1)}^{2} + 1 {(4 a)}^{0} {(- 1)}^{3}$

Evaluate the exponent.

$64 a^{3} + 48 a^{2} \cdot - 1 + 3 {(4 a)}^{1} {(- 1)}^{2} + 1 {(4 a)}^{0} {(- 1)}^{3}$

Multiply $- 1$ by $48$ .

$64 a^{3} - 48 a^{2} + 3 {(4 a)}^{1} {(- 1)}^{2} + 1 {(4 a)}^{0} {(- 1)}^{3}$

Simplify.

$64 a^{3} - 48 a^{2} + 3 (4 a) {(- 1)}^{2} + 1 {(4 a)}^{0} {(- 1)}^{3}$

Multiply $4$ by $3$ .

$64 a^{3} - 48 a^{2} + 12 a {(- 1)}^{2} + 1 {(4 a)}^{0} {(- 1)}^{3}$

Raise $- 1$ to the power of $2$ .

$64 a^{3} - 48 a^{2} + 12 a \cdot 1 + 1 {(4 a)}^{0} {(- 1)}^{3}$

Multiply $12$ by $1$ .

$64 a^{3} - 48 a^{2} + 12 a + 1 {(4 a)}^{0} {(- 1)}^{3}$

Multiply ${(4 a)}^{0}$ by $1$ .

$64 a^{3} - 48 a^{2} + 12 a + {(4 a)}^{0} {(- 1)}^{3}$

Apply the product rule to $4 a$ .

$64 a^{3} - 48 a^{2} + 12 a + 4^{0} a^{0} {(- 1)}^{3}$

Anything raised to $0$ is $1$ .

$64 a^{3} - 48 a^{2} + 12 a + 1 a^{0} {(- 1)}^{3}$

Multiply $a^{0}$ by $1$ .

$64 a^{3} - 48 a^{2} + 12 a + a^{0} {(- 1)}^{3}$

Anything raised to $0$ is $1$ .

$64 a^{3} - 48 a^{2} + 12 a + 1 {(- 1)}^{3}$

Multiply ${(- 1)}^{3}$ by $1$ .

$64 a^{3} - 48 a^{2} + 12 a + {(- 1)}^{3}$

Raise $- 1$ to the power of $3$ .

$64 a^{3} - 48 a^{2} + 12 a - 1$

Expand Using Pascal's Triangle (4a-1)^3

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